Chapter 7: Q. 72 (page 593)

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$\{\frac{(2 k)!}{{(k!)}^{2}}\}$

Short Answer

Expert verified

The given sequence is monotonic and unbounded.

Step by step solution

Step 1. Given Information.

The given sequence is $\{\frac{(2 k)!}{{(k!)}^{2}}\} .$

Step 2. Determine whether the sequences are monotonic or not.

To determine whether the sequences are monotonic or not, we will use the ratio test.

Let the general term of the sequence is $a_{k} = \frac{(2 k)!}{{(k!)}^{2}} .$

So, the $a_{k + 1}$ term is

$a_{k + 1} = \frac{(2 (k + 1))!}{{((k + 1)!)}^{2}} .$

According to the ratio test,

$\frac{a_{k + 1}}{a_{k}} = \frac{\frac{(2 (k + 1))!}{{((k + 1)!)}^{2}}}{\frac{(2 k)!}{{(k!)}^{2}}} = \frac{(2 k + 2)! {(k!)}^{2}}{{((k + 1)!)}^{2} (2 k)!} = {(\frac{k!}{(k + 1)!})}^{2} \cdot (\frac{(2 k + 2)!}{(2 k)!}) = {(\frac{k!}{(k + 1) k!})}^{2} \cdot (\frac{(2 k + 2) (2 k + 1) (2 k)!}{(2 k)!}) = \frac{(2 k + 2) (2 k + 1)}{{(k + 1)}^{2}} = \frac{2 (2 k + 1)}{(k + 1)}$

Now, $\frac{a_{k + 1}}{a_{k}} > 1 for k > 0 .$

Thus, the sequence is strictly increasing and it is monotonic.

Step 3. Determine whether the given sequence is bounded or unbounded.

From the given sequence, we can depict that it is bounded below by zero. Thus, there is no bounded above. Hence the sequence is unbounded.

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Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$\{\frac{(2 k)!}{{(k!)}^{2}}\}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Determine whether the sequences are monotonic or not.

Step 3. Determine whether the given sequence is bounded or unbounded.

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Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.2k!k!2

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Determine whether the sequences are monotonic or not.

Step 3. Determine whether the given sequence is bounded or unbounded.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Probability and Statistics

Decision Maths

Mechanics Maths

Calculus

Pure Maths

Study anywhere. Anytime. Across all devices.

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$\{\frac{(2 k)!}{{(k!)}^{2}}\}$